Mathematics

Mathematics

Problem

John purchases a new bicycle for 880 US dollars (USD) and pays for it with a Canadian credit card. There is a transaction fee of 4.2% charged to John by the credit card company to convert this purchase into Canadian dollars (CAD). The exchange rate is 1 USD = 1.25 CAD. John insures his bicycle with a US company. The insurance company produces the following table for the bicycle's value during each year.
YearValue of the bicycle (USD)
1st880
2nd704
3rd563.20
......
The values of the bicycle form a geometric sequence. During the 1st year John pays 120 USD to insure his bicycle. Each year the amount he pays to insure his bicycle is reduced by 3.50 USD. John purchased the bicycle in 2008. a. Calculate, in CAD, the total amount John pays for the bicycle. b. Find the value of the bicycle during the 5th year. Give your answer to two decimal places. c. Calculate, in years, when the bicycle value will be less than 50 USD. d. Find the total amount John has paid to insure his bicycle for the first 5 years. e. Justify why John should not insure his bicycle in 2019.
Intuitive explanation

The key insight is recognizing that this problem has two independent sequences: the bicycle's value follows a geometric pattern (constant ratio between consecutive terms), while the insurance payments form an arithmetic sequence (constant difference). You'll need to find the common ratio for depreciation and use it to answer questions about the bicycle's value, then separately track the linear decrease in insurance costs. Understanding which sequence applies to which part of the problem prevents mixing up the patterns.

Intuitive explanation

The key insight is recognizing that this problem has two independent sequences: the bicycle's value follows a geometric pattern (constant ratio between consecutive terms), while the insurance payments form an arithmetic sequence (constant difference). You'll need to find the common ratio for depreciation and use it to answer questions about the bicycle's value, then separately track the linear decrease in insurance costs. Understanding which sequence applies to which part of the problem prevents mixing up the patterns.

Browse the problem library